Applications of state space models in finance

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136 7 A Kalman filter based conditional multifactor pricing model annually with the weightings being reviewed quarterly; see Stoxx Ltd. (2005) for more details. As the history of these two indices only goes back to 30 June 1997, V GS is an assembled series: prior to inception of the two DJ Stoxx style indices, the value-growthspread is proxied by the difference between the log-returns of the Frank Russell 1000 Value and the Frank Russell 1000 Growth index. Comparable to the DJ Stoxx style indices for European stock markets, the objective of the two Frank Russell indices is to give a comprehensive picture of value and growth stocks for a universe of the 1000 U.S. common stocks with the highest market capitalization. The style index membership is determined by ranking each stock in the Russell 1000 universe by their respective price to book ratio and their projected long-term growth. The indices are reconstituted on a yearly basis; more details on the methodology are provided by Russell Investment Group (2007). Following this procedure, the value and growth indices, from which the time series of the V GS factor is derived, refer to different regions. However, during the period for which weekly style returns are available for both European and U.S. markets, the correlation between the respective value (growth) returns is as high as 0.76 (0.73). Therefore, under the assumption that style performance in different regions — particularly in the U.S. and Europe — is highly correlated, the chosen procedure can be considered feasible. 7.3.3 The market factor Given the importance of the market portfolio as a measure of aggregate wealth as indicated by the CAPM (cf. ¢ 6), a market factor is typically included as a central risk factor in the context of a multifactor pricing model. Even though the goal of this chapter is to analyze the time-varying impact of macroeconomic and fundamental factors on European industry portfolios, the consideration of a non-macroeconomic, non-fundamental equity market factor is economically plausible: as stock prices respond very quickly to new information, aggregate market returns can be expected to reflect additional information that is not yet captured by the factors above (cf. Chen et al. 1986). As illustrated in Table 7.2, the excess log-returns of the DJ Stoxx Broad index, which serve as a proxy for the overall market throughout this thesis, are highly correlated with some of the chosen macroeconomic and fundamental risk factors. A common approach to assure that the set of considered risk factors is reasonably independent is to define a so-called residual market factor. This factor measures market returns after eliminating the effects of the macroeconomic and fundamental factors. Brennan et al. (2000), for example, regress the series of market returns on the set of the remaining risk factors by using standard OLS. The obtained disturbance terms are then used as the residual market factor. As a means to avoid any look-ahead bias, in this chapter the residual benchmark factor, BMR, is derived as the series of one-step ahead prediction errors. These are obtained from an auxiliary regression with time-varying coefficients modeled as individual random ¢ walks (cf. 3.6.1.1). In order to use the Kalman filter for estimation purposes, the
7.3 The risk factors 137 auxiliary regression is stated in state space form with observation equation R0,t = β01,tV GSt + β02,tSIZt + β03,tT St + β04,tOILt + β05,tF Xt + ɛ0,t, ɛ0,t ∼ N(0, σ2 0 ), (7.15) where R0,t denotes excess returns on the DJ Stoxx of five state equations is given by Broad benchmark index. The set β0k,t+1 = β0k,t + η0k,t, η0k,t ∼ N(0, σ 2 η0k ), (7.16) for k = 1, . . . , 5. In accordance with the definition of one-step ahead prediction errors, v † t , given in (3.83), the residual benchmark series is defined as BMRt := v † t , (7.17) for t = d + k + 1, . . . , T , where d refers to the number of diffuse state elements. In order to receive a series BMR that is of length T, an extended history of 735 observations with starting date 9 January 1991 is employed for estimating the system of equations (7.15)–(7.16). That way, with t = −51, . . . , 683, the series of one-step-ahead prediction errors obtained in (7.17) contains 735 − k − d elements of which the last 683 are taken as residual market factor series BMR. The estimation results are summarized in Table 7.3. As illustrated by the estimated R 2 , only 57% of the variation of the excess benchmark returns are explained by the chosen macroeconomic and fundamental factors. The remaining unexplained information is captured by the generated BMR series. 7.3.4 Summary statistics Summary statistics for the set of chosen risk factors are given in Table 7.4. As illustrated by the second column, with the exception of the size factor, all reported mean values are not significantly different zero. For the size factor, the null hypothesis of a zero mean can be rejected at the 5% significance level. For reasons of parsimony, without having to construct zero mean factors as defined in (7.2), the first moment assumption made in (7.7) is considered to hold at least approximately. According to the estimated annualized standard deviations, the macroeconomic factors are on average more volatile than the residual benchmark and fundamental factors. In columns 4–9, the risk factors’ autocorrelations for different lags are displayed; the last column shows the test statistic for the Ljung-Box test of no autocorrelation. While the BMR factor displays elevated autocorrelations for the first two lags, the two fundamental factors show elevated autocorrelations at the first three lags. For all three equity market related factors, the Q-test Table 7.2: Correlations between the DJ Stoxx Broad index and the chosen macroeconomic and fundamental risk factors (8.1.1992–2.2.2005). DJ Stoxx V GS SIZ T S OIL F X Broad −0.27 0.56 0.10 0.01 0.40
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Sascha Mergner Applications of Stat
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erschienen im Universitätsverlag G
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Bibliographische Information der De
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Contents List of figures . . . . .
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Contents ix 4.5 Model selection and
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Contents xi 7.5 Concluding remarks
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xiv List of figures 6.14 Histograms
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Notation and conventions The outlin
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xx Used abbreviations and symbols L
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xxii Used abbreviations and symbols
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xxiv Used abbreviations and symbols
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Preface The work on this book began
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Chapter 1 Introduction “Economist
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1.2 Research objectives 3 1.2 Resea
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1.3 Organization of the thesis 5 to
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8 2 Some stylized facts of weekly s
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10 2 Some stylized facts of weekly
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18 3 Linear Gaussian state space mo
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Chapter 4 Markov regime switching M
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4.1 Basic concepts 45 R t (%) 20 10
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4.1 Basic concepts 47 probability d
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4.3 Parameter estimation 49 X 1 X 2
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4.3 Parameter estimation 51 4.3.2.1
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4.4 Forecasting and decoding 53 4.4
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4.4 Forecasting and decoding 55 mod
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Chapter 5 Conditional heteroskedast
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5.1 Autoregressive conditional hete
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5.2 Stochastic volatility 65 or the
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5.2 Stochastic volatility 67 5.2.1.
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5.2 Stochastic volatility 69 likeli
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5.2 Stochastic volatility 71 linear
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5.2 Stochastic volatility 73 3. An
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5.3 Multivariate conditional hetero
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84 6 Time-varying market beta risk
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Page 155 and 156: Chapter 7 A Kalman filter based con
Page 157 and 158: 7.1 Factor modeling 125 uncondition
Page 159 and 160: 7.1 Factor modeling 127 predictabil
Page 161 and 162: 7.1 Factor modeling 129 usefulness,
Page 163 and 164: 7.2 Specification of a conditional
Page 165 and 166: 7.3 The risk factors 133 Table 7.1:
Page 167: 7.3 The risk factors 135 introduced
Page 171 and 172: 7.4 Empirical results 139 Table 7.4
Page 173 and 174: 7.4 Empirical results 141 are consi
Page 177 and 178: 7.4 Empirical results 145 β TS 0.4
Page 185 and 186: 7.4 Empirical results 153 cumulativ
Page 187 and 188: Chapter 8 Conclusion and outlook Th
Page 189 and 190: Conclusion and outlook 157 cannot o
Page 191 and 192: Appendix A Brief review of asset pr
Page 193 and 194: A.3 Alternative asset pricing model
Page 195 and 196: Appendix B Figures
Page 197 and 198: Figures 165 β t β t β t 2.0 1.5
Page 209 and 210: Appendix C Tables
Page 211 and 212: Tables 179 Table C.1 — continued
Page 213 and 214: Tables 181 Table C.3: In-sample mea
Page 215 and 216: Tables 183 Table C.5: Out-of-sample
Page 217 and 218: Tables 185 Table C.7: Parameter est
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Tables 187 Table C.7 — continued
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Tables 189 Table C.8: Out-of-sample
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References Abell, J. D. and T. M. K
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References 193 Bulla, J. (2006). Ap
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References 195 Fabozzi, F. J. and J
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References 197 Harvey, A. C., E. Ru
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References 199 Lie, F., R. Brooks,
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References 201 Shephard, N. (1993).
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State space models play a key role
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